# How to visualize a category (of “combinatorial” maps)

This is a practical and very soft question, with the combinatorial database http://www.findstat.org in mind.

I have a few, around 20, families of combinatorial objects, for example Dyck paths, permutations, perfect matchings, graphs, etc., together with a few, around 200, maps between them. The maps need not be injective or surjective or have any special properties. (Except perhaps that they appear in the literature and are therefore "interesting".)

Some examples of such maps might be the reversal of a Dyck path, various classical maps that send a Dyck path to a 321-avoiding permutation, the map that sends a permutation to its shape under the Robinson-Schensted correspondence, etc.

Thus, we have (a very small) category, which I'd like to visualize.

The aspect that makes this interesting and non-trivial is, that these maps satisfies numerous identities: many maps are involutions, some are idempotent, many maps commute with other maps or are conjugate to other maps, etc. I have all these data.

So, more precisely: I'm looking for a way to visualize (graphically!) these identities.

• Silly remark: All relations between these maps can be drawn as commutative diagrams. For commutativity relations this makes sense somehow, but for idempotent maps it is kind of "over the top". – HeinrichD Dec 28 '16 at 17:02
• Yes. The problem is the amount of the data. So, to clarify: the question is about an economic visual representation. – Martin Rubey Dec 28 '16 at 17:16
• It might be overkill, but you could put all the generating morphisms and relations into Globular and then play around with it. Globular is designed to manipulate higher categories, but it should work just fine for 1-categorical data. Incidentally, do you have any map which takes as input a pair of objects, like, say, a Dyck path and a permutation? This would start using some kind of monoidal structure, which is a sort of higher categorical structure. – Tim Campion Dec 28 '16 at 17:52
• @TimCampion: Except that I have no idea how to use it, sounds interesting! – Martin Rubey Dec 28 '16 at 18:48
• @TimCampion: Could you show how to visualize the identities "reverse o inverse = inverse o complement", "reverse o complement = complement o inverse", etc.? (keeping in mind that I have a list of a few thousand such identities...) – Martin Rubey Dec 28 '16 at 18:58